On the $D(4)$-pairs $\{a, ka\}$ with $k\in \{2,3,6\}$

TL;DR

This paper investigates the structure and extensibility of specific $D(4)$-pairs involving multiples of a number, proving that such quadruples are always regular and providing explicit formulas for their elements.

Contribution

It introduces a detailed analysis of $D(4)$-pairs with $b=ka$ for $k ext{ in } oxed{2,3,6}$, establishing conditions for their extension to quadruples and deriving explicit formulas.

Findings

Any $D(4)$-quadruple containing $oxed{a, ka}$ is regular.

Explicit formula for the element $d$ in the quadruple involving $a, b, c$.

Proof that such quadruples are uniquely determined by the given parameters.

Abstract

Let $a$ and $b=ka$ be positive integers with $k\in \{2, 3, 6\},$ such that $ab+4$ is a perfect square. In this paper, we study the extensibility of the $D(4)$-pairs $\{a, ka\}.$ More precisely, we prove that by considering three families of positive integers $c$ depending on $a,$ if $\{a, b, c, d\}$ is the set of positive integers which has the property that the product of any two of its elements increased by $4$ is a perfect square, then $d$ in given by $$d=a+b+c+\frac{1}{2}\left(abc\pm \sqrt{(ab+4)(ac+4)(bc+4)}\right).$$ As a corollary, we prove that any $D(4)$-quadruple which contains the pair $\{a, ka\}$ is regular.